Returns to scale in diﬀerent DEA models
Rajiv D. Banker a, William W. Cooper b, Lawrence M. Seiford c, Robert M. Thrall d, Joe Zhu e,* c
School of Management, The University of Texas at Dallas, Richardson, TX 75083-0658, USA Graduate School of Business, The University of Texas at Austin, Austin, TX 78712-1174, USA Department of Industrial and Operations Engineering, University of Michigan, Ann Arbor, MI 48109-2117, USA d 12003 Pebble Hill Drive, Houston, TX 77024, USA e Department of Management, Worcester Polytechnic Institute, Worcester, MA 01609, USA b
Abstract This paper discusses returns to scale (RTS) in data envelopment analysis (DEA) for each of the presently available types of models. The BCC and CCR models are treated in input oriented forms while the multiplicative model is treated in output oriented form. (This distinction is not pertinent for the additive model which simultaneously maximizes outputs and minimizes inputs in the sense of a vector optimization.) Quantitative estimates in the form of scale elasticities are treated in the context of multiplicative models, but the bulk of the discussion is conﬁned to qualitative characterizations such as whether RTS is identiﬁed as increasing, decreasing or constant. This is discussed for each type of model and relations between the results for the diﬀerent models are established. The opening section describes and delimits approaches to be examined. The concluding section outlines further opportunities for research. Ó 2003 Elsevier B.V. All rights reserved. Keywords: DEA; Eﬃciency; RTS
1. Introduction It has long been recognized that data envelopment analysis (DEA) by its use of mathematical programming is particularly adept at estimating ineﬃciencies in multiple input and multiple output production correspondences. Following Charnes, Cooper and Rhodes (CCR, 1978), a number of diﬀerent DEA models have now appeared in the literature (Seiford and Thrall, 1990; Seiford, 1996).
Corresponding author. Tel.: +1-508-831-5467; fax: +1-508831-5720. E-mail address: email@example.com (J. Zhu).
During this period of model development, the economic concept of returns to scale (RTS) has also been widely studied within the framework of DEA and this, in turn, further extended the applicability of DEA. Two paths may be followed in treating returns to scale (RTS) in DEA. The ﬁrst path, developed by F€re, Grosskopf and Lovell (FGL, 1985, 1994), a determines RTS by a use of ratios of radial measures. These ratios are developed from model pairs which diﬀer only in whether conditions of convexity and sub-convexity are satisﬁed. The second path stems from work by Banker (1984), Banker et al. (1984), Banker and Thrall (1992) and Banker and Maindiratta (1986). This path, which is the
0377-2217/$ - see front matter Ó 2003 Elsevier B.V. All rights reserved. doi:10.1016/S0377-2217(03)00174-7
R.D. Banker et al. / European Journal of Operational Research 154 (2004) 345–362
one we follow, includes, but is not restricted to, radial measure models. It extends to additive and multiplicative models as well, and does so in ways that provide opportunities for added insight into the nature of RTS and its treatment by the methods and concepts of DEA. We restrict attention to the latter and justify our restriction to this (second) alternative by noting that its paths of development have taken a variety of forms that are scattered in the literature. Hence we think the time is ripe to attempt to provide a common source from which further developments may be conveniently essayed. The alternative provided by the FGL approach is an important one, to be sure, so we will take the opportunity to comment further on it in our subsequent discussion. We do not undertake its development in detail in the present paper, however, because we believe that this approach has achieved a considerable...